Finite Group Databasebeta

← #34201677

Subgroups — #34201677

Order 20224.

356 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23765481C212yesmaximal, complemented
2#23763016C211yesmaximal, characteristic, complemented
3#23761439C211yesmaximal, characteristic, complemented
4#23761373C212yesmaximal, complemented
5C79×#3293C211yesmaximal, characteristic
6#19099003—22no—
7#19098966—22no—
8#19098943C2211yescharacteristic
9#19098384C2212yes—
10#19098378—22no—
11C8×C79⋊C8C2211yescharacteristic
12#578×C79—22no—
13#494×C79C2211yescharacteristic
14#473×C79—22no—
15#456×C79—22no—
16#443×C79C2212yes—
17C2×Dic316—22no—
18#14579161—21no—
19#14579107—42no—
20#14579105—42no—
21#14579101—22no—
22#14579094—44no—
23C4×C79⋊C8—21no—
242−1+4×C79—22no—
25#137×C79—24no—
26#137×C79—44no—
27#135×C79—42no—
28#135×C79D412yes—
29C2×Q16×C79—22no—
30C2×Q16×C79—42no—
31#131×C79—22no—
32C4⋊Q8×C79D412yes—
33#108×C79D412yes—
34#104×C79—22no—
35#101×C79—22no—
36C4×C8×C79C2311yescharacteristic
37C2×Dic158—41no—
38Dic316—42no—
39Dic316—42no—
40C2×C79⋊C8—22no—
41C4⋊Dic79—42no—
42C79⋊C16—42no—
43#41×C79—22no—
44#41×C79—44no—
45#41×C79—82no—
46C2×Q8×C79—22no—
47C2×Q8×C79—44no—
48Q16×C79—44no—
49Q16×C79—42no—
50Q16×C79—42no—
51Q16×C79D812yes—
52SD16×C79—22no—
53SD16×C79—44no—
54D8×C79D812yes—
55M16×C79—22no—
56M16×C79—42no—
57C2×C8×C79—42no—
58C2×C8×C79C2×D412yes—
59C4⋊C4×C79—42no—
60C42×C79C2×D411yescharacteristic
61Q8×C79—22no—
62Q8×C79—44no—
63Q8×C79—82no—
64D4×C79—22no—
65D4×C79—42no—
66D4×C79—82no—
67C2×C4×C79—21no—
68C2×C4×C79—42no—
69C2×C4×C79—42no—
70C2×C4×C79—84no—
71C2×C4×C79#12011yescharacteristic
72C2×Dic79—81no—
73Dic158—82no—
74C8×C79—42no—
75C8×C79C2×D812yes—
76C8×C79AGL(1,ℤ/8)12yes—
77C79⋊C8—24no—
78C22×C79—42no—
79C22×C79#45311yescharacteristic
80C4×C79—22no—
81C4×C79—42no—
82C4×C79—82no—
83C4×C79#44712yes—
84Dic79—161no—
85#12313—791nomaximal, Sylow
86C2×C79—21no—
87C2×C79—82no—
88C2×C79#161911yescharacteristic
89#3293D7911yescharacteristic
90#2214—792no—
91#1713—791no—
92#1358—791no—
93#1335—792no—
94C79#1231311yescharacteristic, complemented, Sylow
95#578—22no—
96#510—1582no—
97#500—1582no—
98#494—791no—
99#494C2×D7911yescharacteristic
100#473—22no—
101#456—22no—
102#443C2×D7912yes—
103#364—792no—
104#362—1582no—
105C82—791no—
1062−1+4—22no—
107#137—24no—
108#137—44no—
109#135—42no—
110#135C22⋊D7912yes—
111C2×Q16—22no—
112C2×Q16—42no—
113C2×Q16—1582no—
114#131—22no—
115C4⋊Q8—1581no—
116C4⋊Q8C22⋊D7912yes—
117Q32—3162no—
118SD32—3162no—
119M32—1582no—
120#108C22⋊D7912yes—
121#107—3164no—
122#104—22no—
123#101—22no—
124C4×C8—1581no—
125C4×C8C22×D7911yescharacteristic
126#41—22no—
127#41—44no—
128#41—82no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—3161no—
132Q16—44no—
133Q16—42no—
134Q16—42no—
135Q16—3162no—
136Q16—3162no—
137Q16#1192960912yes—
138SD16—22no—
139SD16—44no—
140D8#1192960912yes—
141M16—22no—
142M16—42no—
143C2×C8—42no—
144C2×C8—1582no—
145C2×C8C2×C22⋊D7912yes—
146C4⋊C4—42no—
147C4⋊C4—3162no—
148C42C2×C22⋊D7911yescharacteristic
149C16—3162no—
150Q8—22no—
151Q8—44no—
152Q8—82no—
153Q8—6322no—
154D4—22no—
155D4—42no—
156D4—82no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4—6321no—
162C2×C4#1457913011yescharacteristic
163C8—42no—
164C8—1584no—
165C8C2×#1192960912yes—
166C8#1457918712yes—
167C22—42no—
168C22#1909873111yescharacteristic
169C4—22no—
170C4—42no—
171C4—82no—
172C4—12641no—
173C4#1909869712yes—
174C2—21no—
175C2—82no—
176C2#2376245411yescharacteristic
subgroupG/H
centerC2#23762454
commutator, cosocleC4×C8×C79C23
FrattiniC4×C8C22×D79
Fitting, prime radicalC79×#3293C2
socleC2×C79#1619
hypercenter#3293D79
hypocenterC79#12313
LCSG ▷ C4×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3293

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23765481C2semidirect
2. #23763016C2semidirect
3. #23761439C2semidirect
4. #23761373C2semidirect
94. C79#12313semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237654812
#237630162
#237614392
#237613732
C79×#32932
#1231379